Question: Simplify the following expression and state the condition under which the simplification is valid. $r = \dfrac{3q^3 - 15q^2 - 150q}{-2q^2 + 26q - 60}$
First factor out the greatest common factors in the numerator and in the denominator. $ r = \dfrac {3q(q^2 - 5q - 50)} {-2(q^2 - 13q + 30)} $ $ r = -\dfrac{3q}{2} \cdot \dfrac{q^2 - 5q - 50}{q^2 - 13q + 30} $ Next factor the numerator and denominator. $ r = - \dfrac{3q}{2} \cdot \dfrac{(q - 10)(q + 5)}{(q - 10)(q - 3)}$ Assuming $q \neq 10$ , we can cancel the $q - 10$ $ r = - \dfrac{3q}{2} \cdot \dfrac{q + 5}{q - 3}$ Therefore: $ r = \dfrac{ -3q(q + 5)}{ 2(q - 3)}$, $q \neq 10$